Some of your trouble may come from ingredients.
Let's start with buttercream. It has to be soft enough to spread, and that means a useful amount of liquid.
Your sugar has to be proper 10X powdered--stick your hand into it rub it with your fingers. It should be powdery and silky. If it's gritty at all then you have 6X not 10X. FYI the Domino 25lb bags are 6X according to the Domino website. Bulk pack of "house" brands may also be 6X.
Go back to basics: make some icing using 1 pound no-trans-fat crisco, 1 pound salted butter taken from the fridge 4 hours before starting, 4 pounds 10X sugar, and 1 cup of liquid. For liquid, pre-mix 3/4 cup dry Carnation Coffee-mate and 3/4 cup hot water immediately before you are going to start mixing the icing. After mixing, measure out 1 cup for the icing. Also 2 tablespoons real vanilla extract. Warm the liquid in the microwave for 20 seconds immediately before you add it.
Mix this by beating the butter and crisco on medium speed for a few minutes. Add the sugar 1 cup at a time and beat until smooth each time. Yes this will take 10 minutes. That's part of the secret. Turn the mixer to high speed and add the vanilla, and then the HOT liquid 2 tablespoons at a time. Beat until fluffy before adding more liquid. If the icing has not become fluffy, warm up and add the rest of the liquid that you made.
Use this icing to practise making a smoothly iced cake. Never mind the decorating, anytime you bake cake at home, use this on it. Or use just a cake tin...Store it in the freezer, bring it back to room temperature, and re-beat it before using it. It has to be soft enough to spread properly and that means it has to be at room temperature.
Finally getting good corners on buttercream means that the cake has been cut level, and then you put on a layer of icing AT LEAST 1/4" thick. You would do better with 3/8" to learn. You then use a long bakers spatula NOT offset to remove icing until the corners are sharp. No need for paper towels...
And you know, the classic way of covering cakes with fondant has been with NO sharp corners for at least two centuries. The only icing that used to be iced "sharp" was royal icing on fruitcakes with dead flat marzipan undercoat, and they laid at least 3 layers of royal icing...
That's how I always baked for customers: buttercream gets sharp corners, and fondant does NOT.